Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension

Sims, Aidan; Szabó, Gábor; Williams, Dana P.

doi:10.1007/978-3-030-39713-5

Cited by 63 publications

(93 citation statements)

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“…(4) Let X be a G ′ -set, G ′ acts on X. The transformation groupoid G is the category with Ob(G) = X and G(x, y) = {g ∈ G ′ : g • x = y} (see example 8.1.15 in [25]). Let P be the wide subcategory of G such that P(x, y) = {g ∈ P ′ : g • x = y}.…”

Section: Product Systems Over Quasi-lattice Ordered Groupoidsmentioning

confidence: 99%

“…The reduced groupoid C * -algebra C * r (G) of G is the C * -algebra generated by {λ g } g∈G(−,−) (see [25]). The full groupoid C * -algebra C * (G) is the universal C * -algebra generated by the representation of G. In particular, there is a * -homomorphism from C * (G) to C * r (G) defined by U g → λ g , where U g is the generator of C * (G) associated with g (see Chapter 9 in [25]).…”

Section: Introductionmentioning

confidence: 99%

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Co-universal $C^{\ast}$-algebras for product systems over quasi-lattice ordered groupoids

Feifei¹,

Wang²,

Wang³

2022

Preprint

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We introduce the notion of product systems over quasi-lattice ordered groupoids, and characterize the co-universal algebras for compactly aligned product systems over quasi-lattice ordered groupoids by using the C * -envelopes of the cosystems associated with the product systems.2010 Mathematics Subject Classification. 46L05. Key words and phrases. Co-universal C * -algebras; Product systems; Quasi-lattice ordered groupoids.

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Section: Product Systems Over Quasi-lattice Ordered Groupoidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Co-universal $C^{\ast}$-algebras for product systems over quasi-lattice ordered groupoids

Feifei¹,

Wang²,

Wang³

2022

Preprint

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“…In Section 2, we follow [7,Chapter 2] and construct a secondcountable, amenable, locally compact, Hausdorff andétale groupoid G X whose C *algebra is canonically isomorphic to O X . For an introduction to (étale) groupoid C * -algebras see [46,49] or the introductory notes [52].…”

Section: Preliminariesmentioning

confidence: 99%

𝐶*-algebras, groupoids and covers of shift spaces

Brix¹,

Carlsen²

2020

Trans. Amer. Math. Soc. Ser. B

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To every one-sided shift space X we associate a cover X, a groupoid G X and a C *-algebra O X. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between X and Y in terms of isomorphism of G X and G Y , and diagonal-preserving *-isomorphism of O X and O Y. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces Λ X and Λ Y in terms of isomorphism of the stabilized groupoids G X × R and G Y × R, and diagonal-preserving *-isomorphism of the stabilized C *-algebras O X ⊗ K and O Y ⊗ K. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of X from the pair (O X , C(X)), and the flow equivalence class of Λ X from the pair (O X ⊗ K, C(X) ⊗ c 0). In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces. Contents 154 7. Two-sided conjugacy 166 8. Flow equivalence 170 References 182

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“…In this section we recall the notions ofétale groupoids and groupoid C*-algebras. We refer to [9,10,12] for details.…”

Section: éTale Groupoids and Groupoid C*-algebrasmentioning

confidence: 99%

“…Proof. Recall that anétale groupoid G is Hausdorff if and only if its unit space G (0) is a closed subset of G (see, for example, [12…”

Section: Quotients Ofétale Groupoidsmentioning

confidence: 99%

Quotients of Étale Groupoids and the Abelianizations of Groupoid C*-Algebras

Komura

2020

J. Aust. Math. Soc.

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Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension

Cited by 63 publications

References 0 publications

Co-universal $C^{\ast}$-algebras for product systems over quasi-lattice ordered groupoids

Co-universal $C^{\ast}$-algebras for product systems over quasi-lattice ordered groupoids

𝐶*-algebras, groupoids and covers of shift spaces

Quotients of Étale Groupoids and the Abelianizations of Groupoid C*-Algebras

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